What is the Newton-Raphson Method?

Newton's Method May Fail

This method may fail under two conditions:

• The function's first derivative evaluated at the assumed value is zero, \(f^{\prime}(x_1)=0\). If we look at the general expression, the denominator \(f^{\prime}(x)\) cannot be zero (division by zero is impossible).
• After performing multiple iterations, the computed values diverge from each other (doesn't meet on a singular point).

Example

Say we want to find the zeroes of \(x^2+2x-4=0\). From the degree of the polynomial function, there would be two zeroes.

To start, we need to assume a zero value of the function. We begin with \(x=2\) and perform the above steps. Eventually, it will converge to a zero equal to 1.236...

\(x_{n+1}=x_{n}-\frac{f(x_{n})}{f^\prime(x_{n})}\)

\(x_{n+1}=x_{n}-\frac{x_{n}^2+2x_{n}-4}{2x_{n}+2}\)

First Iteration

• \(x_2=2-\frac{(2)^2+2(2)-4}{2(2)+2}\)
• \(x_2=1.333...\)

Second Iteration

• \(x_2=1.33-\frac{(1.33)^2+2(1.33)-4}{2(1.33)+2}\)
• \(x_2=1.238...\)

Third Iteration

• \(x_2=1.238-\frac{(1.238)^2+2(1.238)-4}{2(1.238)+2}\)
• \(x_2=1.236...\)

To find the other zero, we assume another value. Say, we start with \(x=-2\). Performing the steps above, we will have a zero of -3.236...

Applications

Now that we know the general procedure, where do we see this method? A great example would be scientific calculators. Most of them solve for the zeroes of the function using this way. 

• Input a function expression in the calculator.
• When we try to solve it, the calculator will ask for an \(x\) value first. It is the starting point of the calculation.
• The calculator will perform the necessary procedure behind the scenes.
• Afterward, it will give the zero of the function.

Summary

The Newton-Raphson Method (or simply Newton's Method) is a way to approximate the zeroes of the function. We can use it as an alternative to solve functions or equations. 

We can summarise this method with the general formula: \(x_{n+1}=x_n-\frac{f\left(x_n\right)}{f^{\prime}\left(x_n\right)}\). \(x_{n+1}\) is the x-value of the next coordinate, \(x_n\) is the x-value of the original coordinate, \(f\left(x_n\right)\) is the evaluated function with \(x_n\), and \(f^{\prime}\left(x_n\right)\) is the evaluated first derivative of the function with \(x_n\)

We first assume an \(x_n\) value and repeatedly use the formula to arrive at zero.

The more repetitions we make with the formula, the more accurate the result is. It will eventually converge to a singular x-point, the function's zero.

This method may fail under two conditions: (1) the function's first derivative evaluated at the assumed value is zero, \(f^{\prime}(x_1)=0\), and (2) after performing multiple iterations, the computed values diverge from each other.